Question 3
Let $n$ be a positive integer and suppose that $n = a_1 + a_2 + \cdots + a_m$ for $m \ge 2$ and $a_1 \gt a_2 \gt \cdots \gt a_m$ distinct positive integers satisfying
$$
\frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_m} = 1.
$$
Prove that the inequality $n + 2m + 1 \le a_1 a_2$ holds and determine the conditions under which equality holds.
$$
\frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_m} = 1.
$$
Prove that the inequality $n + 2m + 1 \le a_1 a_2$ holds and determine the conditions under which equality holds.